Given topological spaces $X_1, X_2, \dotsc, X_n, Y$, consider a multivariable function $f : \prod_{i = 1}^nX_i \to Y$ such that for any $(x_1, x_2, \dotsc, x_n) \in \prod_{i = 1}^nX_i$, the functions in the family $\{x \mapsto f(x_1, \dotsc, x_{i - 1}, x, x_{i + 1}, \dotsc, x_n)\}_{i = 1}^n$ are all continuous. Must $f$ itself be continuous? It seems to be true, so is there a Theorem that proves this?
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The answer is "Yes" if you give the product $\prod X_i$ the slice (or sometimes called "cross topology"). The answer is "No" if you give the product $\prod X_i$ the usual product topology. In this case, $f$ is "separately continuous" but not necessarily continuous. The standard example is the function $f:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ defined by $f(x,y)=\frac{2xy}{x^2+y^2}$ for $(x,y)\neq (0,0)$ and $f(0,0)=(0,0)$. This function is continuous everywhere except $(0,0)$ but is continuous in each variable. |
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